4.6 Related Rates
What are related rates problems?
• If several variables that are functions of time t
are related by an equation, we can obtain a
relation involving their time rates by
differentiating with respect to t.
General Process for Solving Related Rate Problems
1. Draw a diagram.
2. Represent the given information and the unknowns by
mathematical symbols.
3. Write an equation involving the rate of change to be
determined. (If the equation contains more than one
variable, it may be necessary to reduce the equation to
one variable.)
4. Differentiate each term with respect to time (t). (you
will use a form of implicit differentiation)
5. Substitute all known values and know rates of change
into the resulting equation.
6. Solve the resulting equation for the desired rate of
change.
7. Write the answer with units of measure.
Example
• If one leg AB of a right triangle increases at the rate of 2
inches per second, while the other leg AC decreases at
3 inches per second, find how fast the hypotenuse is
changing when AB = 72 inches and AC = 96 inches.
C
y
x
A
Given:
dz
 2 in
sec
dt
z
z  72 in.
B
dx
 3 in
sec
dt
x  96 in.
dy
Find
dt
x 2  z 2  y 2 Differentiate with respect to time.
dx
dz
dy
2x  2z
 2y
Divide out a “2”.
dt
dt
dt
Example
• If one leg AB of a right triangle increases at the rate of 2
inches per second, while the other leg AC decreases at
3 inches per second, find how fast the hypotenuse is
changing when AB = 72 inches and AC = 96 inches.
C
y
x
A
z
x2  z 2  y2
(96)  (72)  y
2
120  y
Given:
dz
 2 in
sec
dt
2
z  72 in.
B
dx
 3 in
sec
dt
x  96 in.
dx
dz
dy
x z
y
dt
dt
dt
dy
96(3)  72(2)  y
dt
dy
Find
dt
Plug in known values.
What is y?
Example
• If one leg AB of a right triangle increases at the rate of 2
inches per second, while the other leg AC decreases at
3 inches per second, find how fast the hypotenuse is
changing when AB = 72 inches and AC = 96 inches.
C
y
x
A
Given:
dz
 2 in
sec
dt
z
z  72 in.
B
dx
 3 in
sec
dt
x  96 in.
dy
96(3)  72(2)  120
dt
 1.2 in
dy

sec dt
dy
Find
dt
Solve for dy/dt
Example
• The diameter and height of a paper cup in the shape of a
cone are both 4 inches, and water is leaking out at the
rate of ½ cubic inch per second. Find the rate at which
the water level is dropping when the diameter of the
surface is 2 inches.
Example
• A bouillon cube with side length 0.8 cm is placed into
boiling water. Assuming it roughly resembles a cube as
it dissolves, at approximately what rate is its volume
changing when its side length is 0.25 cm and is
decreasing at a rate of 0.12 cm/sec?
Example
• A 20-foot extension ladder propped up against the side
of a house is not properly secured, causing the bottom of
the ladder to slide away from the house at a constant
rate of 2 ft/sec. How quickly is the top of the ladder
falling at the exact moment the base of the ladder is 12
feet away from the house?